This example shows how to estimate regularized ARX models using automatically generated regularization constants in the System Identification app.
filename = fullfile(matlabroot,'help','toolbox',... 'ident','examples','ex_arxregul.sid'); systemIdentification(filename)
The session imports the following data and model into the System Identification app:
The data is collected by simulating a system with the following known transfer function:
Transfer function model
trueSys is the transfer function model used to generate the
eData described previously. You also use the impulse
response of this model later to compare the impulse responses of estimated ARX models.
In the System Identification app, select Estimate > Polynomial Models to open the Polynomial Models dialog box.
Verify that ARX is selected in the Structure list.
In the Orders field, specify [0 50 0] as the ARX model order and delay.
Click Estimate to estimate the model.
arx0500 is added to the System Identification app.
In the Polynomial Models dialog box, click Regularization.
In the Regularization Options dialog box, select
TC from the
Regularization Kernel drop-down list.
Specifying this option automatically determines regularization constants using the
TC regularization kernel. To learn more, see the
arxRegul reference page.
Click Close to close the dialog box.
In the Name field in the Polynomial Models dialog box, type
arx0500reg is added to the System Identification app.
Select the Model output check box in the System Identification app.
The Measured and simulated model output plot shows that both the models have an 84% fit with the data.
Because the model fit to the estimation data is similar with and without using
regularization, compare the impulse response of the ARX models with the impulse responses of
trueSys, the system used to collect the estimation data.
trueSys icon in the model board of the System
Select the Transient resp check box to open the Transient Response plot window.
By default, the plot shows the step response.
In the Transient response plot window, select Options > Impulse response to change to plot to display the impulse response.
Select Options > Show 99% confidence intervals to plot the confidence intervals.
The plot shows that the impulse response of the unregularized model
arx0500 is far off from the true system and has huge
To get a closer look at the model fits to the data and the variances, magnify a portion of the plot.
The fit of the regularized ARX model
arx0500reg closely matches the
impulse response of the true system and the variance is greatly reduced as compared to the